[Physics FAQ] - [Copyright]

Original by Philip Gibbs 1996.

Can Special Relativity Handle Acceleration?

It's a common misconception that special relativity cannot handle accelerating objects or accelerating reference frames.  Sometimes it's claimed that general relativity is required for these situations, the reason being given that special relativity only applies to inertial frames.  This is not true.  Special relativity treats accelerating frames differently from inertial frames, but can still deal with accelerating frames.  And accelerating objects can be dealt with without even calling upon accelerating frames.

This idea that special relativity cannot handle acceleration or accelerated frames often comes up in the context of the twin paradox, when people claim that it can only be resolved in general relativity because of the acceleration present.  Their claim is wrong.

The only sense in which special relativity is an approximation when there are accelerating bodies is that gravitational effects such as the generation of gravitational waves are being ignored.  But, of course, there are larger gravitational effects being neglected even when massive bodies are not accelerating, and these are small for many applications, so this is not strictly relevant.  Special relativity gives a completely self-consistent description of the mechanics of accelerating bodies neglecting gravitation, just as newtonian mechanics does.

One difference between general and special relativity is that in the general theory all frames of reference, including spinning and accelerating frames, are treated on an equal footing.  In special relativity accelerating frames are different to inertial frames.  Velocities are relative but acceleration is treated as absolute.  In general relativity all motion is relative.  To accommodate this change, general relativity has to use curved space-time.  In special relativity space-time is always flat.

In special relativity an accelerating particle's worldline is not straight.  This isn't difficult to handle.  The particle's 4-vector acceleration can be defined as the derivative with respect to proper time of its 4-velocity.  It is possible to solve the equations of motion for a particle in electric and magnetic fields, for example.

Accelerating reference frames are a different matter. In GR the physical equations take the same form in any co-ordinate system.  In SR they don't, but it's still possible to use co-ordinate systems corresponding to accelerating or rotating frames of reference, just as it is possible to solve ordinary mechanics problems in curvilinear co-ordinate systems.  This is done by introducing a metric tensor.  The formalism is very similar to that of many general relativity problems, but it is still special relativity as long as the space-time is constrained to be flat and minkowskian.  Note that the speed of light is rarely a constant in non-inertial frames, and this has been known to cause confusion.

An example is a rotating frame of reference used to deal with a rotating object.  The transformation of the metric into the rotating frame leads to "fictitious" forces: Coriolis forces and centrifugal forces.  But this is no different from ordinary mechanics.

A simple task is to solve for the motion of a rocket that accelerates "uniformly".  What does this mean?  We don't mean that its acceleration as measured by an inertial observer is constant.  We mean that it is moving such that its acceleration measured in a "momentarily comoving inertial frame" is always the same; this frame is an inertial frame travelling at the same instantaneous velocity as the object at any moment.  If you were on board such a uniformly accelerated rocket, you would experience a constant "G force".  The motion of this rocket can be found in several ways.  One way uses the four-vector acceleration along the rocket's worldline, since this has constant magnitude.  Alternatively, the rocket is passing constantly from one inertial frame to another in such a way that its change of speed in a fixed time interval is always the same.  From our understanding of adding velocities, we can see that the rapidity r of the rocket must be increasing at a constant rate a with respect to the rocket's proper time T.  The rapidity is related to velocity v by the equation

      v = c tanh(r/c)

From this we derive the equation

      v = c tanh(aT/c)

For other acceleration equations see the relativity FAQ article on the relativistic rocket.